SIAM Journal on Applied Mathematics, Volume 81, Issue 2, Page 530-550, January 2021.
Modeling and simulation of neuronal processes has evolved from the one-dimensional (1D), symmetry-exploiting cable equation to detailed 3D models that incorporate the ultrastructural architecture of neurons. While dimension-reduced models have the clear advantage of simplicity, which makes numerical simulation an easy task, they neglect structural details which may be relevant when studying the structure-function interplay in cells. The Poisson--Nernst--Planck equations represent the other end of the modeling spectrum, a continuum mechanics model that accounts for the 3D domain as well as the spatio-temporal ion concentrations involved in biochemical signaling. In the presented work, we show that three simpler models can be derived from the Poisson--Nernst--Planck equations: the broadly used 1D cable equation, a 3D version thereof, and a 1D electro-diffusion model. We identify the assumptions and approximations that are made for the simpler models and, using numerical simulation, assess the impact of violations of these assumptions on model accuracy in scenarios ranging from individual axon fibers and bundles to dendritic spine microdomains. It is shown that the coarsest model (1D cable equation) is able to accurately resolve the electric behavior in all scenarios, as long as the cellular geometry does not diverge significantly from symmetric local cylinders. In cases where the ionic spatio-temporal dynamics are relevant, more detailed models like the 3D Poisson--Nernst--Planck equations need to be employed. We conclude with guidelines as to which level of modeling detail is necessary to capture the underlying neurobiological features accurately.

中文翻译：

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神经元信号传递的必要建模细节：一维和三维的泊松-能斯特-普朗克和电缆方程模型

SIAM应用数学杂志，第81卷，第2期，第530-550页，2021年1月。
神经元过程的建模和仿真已经从一维（1D），利用对称性的电缆方程演化为包含神经元超微结构的详细3D模型。降维模型具有简单性的明显优势，这使数值模拟成为一项轻松的任务，但它们却忽略了结构细节，而这些细节在研究单元格中的结构-功能相互作用时可能是相关的。泊松-能斯特-普朗克方程代表了建模光谱的另一端，这是一个连续的力学模型，说明了3D域以及生化信号中涉及的时空离子浓度。在展示的工作中，我们展示了可以从Poisson-Nernst-Planck方程得出三个更简单的模型：广泛使用的1D电缆方程，其3D版本，一维电扩散模型 我们确定了为简单模型所做的假设和近似，并使用数值模拟评估了从单个轴突纤维和纤维束到树突状脊柱微区的情况，违反这些假设对模型准确性的影响。结果表明，只要单元格的几何形状与对称的局部圆柱体之间的差异不大，最粗糙的模型（一维电缆方程式）就能在所有情况下准确地解析电行为。在离子时空动力学相关的情况下，需要使用更详细的模型，例如3D Poisson-Nernst-Planck方程。我们以准则为准，以准确地捕获基本的神经生物学特征所需的建模细节级别为准。我们确定为简单模型所做的假设和近似，并使用数值模拟，评估从单个轴突纤维和纤维束到树突状脊柱微区的情况，违反这些假设对模型准确性的影响。结果表明，只要单元格的几何形状与对称的局部圆柱体之间的差异不大，最粗糙的模型（一维电缆方程式）就能在所有情况下准确地解析电行为。在离子时空动力学相关的情况下，需要使用更详细的模型，例如3D Poisson-Nernst-Planck方程。我们以准则为准，以准确地捕获基本的神经生物学特征所需的建模细节级别为准。我们确定了为简单模型所做的假设和近似，并使用数值模拟评估了从单个轴突纤维和纤维束到树突状脊柱微区的情况，违反这些假设对模型准确性的影响。结果表明，只要单元格的几何形状与对称的局部圆柱体之间的差异不大，最粗糙的模型（一维电缆方程式）就能在所有情况下准确地解析电行为。在离子时空动力学相关的情况下，需要使用更详细的模型，例如3D Poisson-Nernst-Planck方程。我们以准则为准，以准确地捕获基本的神经生物学特征所需的建模细节级别为准。